A 1000N black is to be lifted using a chain sling (as shown). If the tension in the chain sling is not to exceed 1300N, find the shortest sling ABC that can be used.
A is the most left point, B is the most right point and C is the top point.

I think i have to take the derivative of something to find the minimum but have no idea how to go about that.

No derivatives needed at all - this problem is easier than you think. To lift the bar the vertical component of force applied by the chain at each end is 1/2 the bar's weight, or W/2. But there is also a horizontal component of force as well, since a chain lifts by tension and tension acts only in the direction of the chain. If T is the tension in the chain, and theta is the angle of the chain to the bar, then

T sin(theta) = W/2

From geometry to get the shortest chain possible you want theta to be small. But note that as theta goes from large to small values the value of T gets bigger. You've been given a max value of T, so now you can determine what the corresponding value for theta is, and from that the length of the chain.